쉐도잉 연습: How real are the real numbers, really? - 영상으로 영어 말하기 배우기

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Say, Achilles wants to run a race.

First he'd have to run half the way, and then half the rest of the way, and half of that, and so on.

But no, this video isn't about Xeno's paradox.

I just wanted to point out, to get to the end, Achilles had to run through every fraction of the length at some point.

So here's the question for the video.

Does Achilles just run from fraction to fraction.

If your first instinct is like mine, you might say of course not.

He doesn't jump from one spot to another.

He smoothly goes along all the points.

But see, if he does just go along all the fractions, he doesn't have to jump.

He starts at 0, but what's the closest fraction to 0?

Well, pick any really small fraction, and there's always one that's closer.

Just pick the one that's half of that.

Then pick one that's closer by looking at the halfway point again and again and again to infinity.

This means that there's never any distance between the fraction Achilles is on

and the next one and so he never has any jumps.

So it looks like we're done.

Achilles can move smoothly along all the fractions.

But see how we've introduced infinity here?

The total distance is just a finite number, but for Achilles to move smoothly along it, he has to go to every fraction,

and there are a lot of fractions.

It's not just that there is an infinite number of fractions along this line, but that between any two fractions, no matter how tiny the distance is between them,

there is an infinite number more fractions.

That's an insane infinity.

But don't worry, there is way way more.

When we think of writing a fraction out in decimal, it's either some finite string of digits or sometimes an infinite one.

But not all infinite strings of numbers are fractions.

Of course, there are all the classic examples.

We'll call these numbers irrational, the ones that can't be written as fractions but can be written as huge strings of numbers.

We said that the fractions have no gaps, right?

But consider this.

Pick some irrational number, any.

Let's say mine starts off like this.

Then say Achilles runs with a constant speed, but at first he runs 0 .7 of the way, which is a fraction, so he can do that.

Then he runs 0 .09 more, and again, that's a fraction, so it's fine.

Then 0 .003, etc, etc. etc. Each bit of this race is a fraction and so he's allowed to run that much,

but he'll stop when he reaches the end of this process.

Now you might be saying he never will because there's an infinite number of steps, but don't forget he's running at a constant speed,

that means the amount of time it takes him to run all these smaller steps is tiny.

So he'll get there in a finite amount of time, in fact in slightly less time than it would take him to run 0 .8 of the way.

So he gets there alright.

But the question is, where is there?

It's this irrational number.

If he could only land on fractions, he would be fine throughout this whole process since he was landing on fractions.

But he gets to the end and it's this point we didn't think existed.

So does he just disappear?

Since that doesn't seem very reasonable, we're forced to accept that all of these irrational numbers are there too,

between the fractions, even though we showed that there is no gaps between the fractions.

The fractions and the irrationals are collectively called the real numbers because they're the ones we need for the real world.

The fact that we need real numbers might seem obvious now, but it was a long, long process in maths to invent the right numbers to support reality.

This problem really puzzled mathematicians.

Suppose we want to know how fast Achilles is running.

If we want his average we just get the total distance and divide it by his time.

But say Achilles tends to change his speed quite a lot during the race.

We want to know exactly what his speed was at a particular point.

Then what do we do?

A better estimate is to start timing for a short amount of time after he's passed that point, and see how far he's gone then.

Then find the average speed from that.

But of course, that's not completely accurate either.

His speed may have changed a bit even in that smaller time.

You'd want to make it even more accurate by picking a smaller time still and doing the same thing.

But of course, no matter how small a time you pick, this is never 100 % accurate.

But we would kind of like to know his speed at this point in order to do physics.

Well then, Leibniz and Newton came along with a great idea.

They said why don't you look at what happens when that time gap is really small.

Like basically zero, but not zero, but smaller than any number.

Let's call it an infinitesimal.

Then do the same calculation with your infinitesimal as your time.

Done.

Problem solved.

Now if that sounds like magic to you, then don't worry, it is.

But they cheerfully accepted dividing by zero because, well, it worked.

But it eventually started to really bother mathematicians.

They didn't like these weird infinitesimals and how they weren't zero sometimes but they were later.

I mean, it's not very rigorous and mathematical sounding, is it?

So they went and started upheaving big chunks of maths to start making it more rigorous,

in the hope that they would eventually be able to explain calculus without the magic.

Their hunt led them all the way back to the numbers themselves.

They realised that what they needed was the real numbers.

See, this property that the real numbers had, that if a sequence is getting closer and closer together, then it is going to converge to something,

That's exactly what you need.

Let's see why.

Back to our speed calculation.

If you do that process of averaging, each time halving the time that you use, and repeat to infinity, you get a long sequence of average times.

Let's say something like this.

Then you can see that the numbers are getting closer.

But since we're dealing with the real numbers, we know that this is going towards some number number.

And whatever number that is, define that to be the speed of Achilles at this point.

So real numbers saved the day, but not without a price.

We saw that even in a finite amount of space there is an infinite number of fractions

and then an infinite number of irrationals slotted between them.

That is much more infinity than you might have imagined fitting in there.

But the plot thickens.

See, mathematicians were very happy to discard all those nasty infinitesimals from maths.

But in the 60s, a mathematician discovered that you can reformulate all of calculus completely rigorously using infinitesimals.

Using that method, the dividing by zero but not really dividing by zero thing, is completely valid.

But instead of getting completely rid of real numbers to do this, you actually add in another infinite set of numbers,

called the hyperreels, that slot into the gaps between the reals.

But that is a story for another time.

100

The conclusion of this video is that there is an infinite number of numbers in the universe.

101

But is the entire universe infinite, or does it have an edge?

102

Answering that is going to take some astrophysics and some general relativity.

103

That sounds really interesting.

104

You should go check out Nick's video.

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쉐도잉 연습: How real are the real numbers, really? - 영상으로 영어 말하기 배우기

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